On a degenerate elliptic problem arising in the least action principle for Rayleigh-Taylor subsolutions

被引：1
作者：
Gebhard, Bjorn [1 ,2 ]
Hirsch, Jonas [3 ]
Kolumban, Jozsef J. [4 ,5 ]
机构：
[1] Univ Autonoma Madrid, Dept Matemat, Madrid 28049, Spain
[2] ICMAT, C Nicolas Cabrera 13-15, Madrid 28049, Spain
[3] Univ Leipzig, Math Inst, Augustuspl 10, D-04109 Leipzig, Germany
[4] Budapest Univ Technol & Econ, Dept Anal & Operat Res, Muegyet Rkp 3, H-1111 Budapest, Hungary
[5] HUN REN Alfred Renyi Inst Math, Realtanoda Utca 13-15, H-1053 Budapest, Hungary
来源：
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE | 2024年 / 41卷 / 06期
关键词：
least action principle; degenerate elliptic variational problem; Euler equations; Boussinesq equation; Rayleigh-Taylor instability; subsolutions; generalized flows; convex integration; EULER EQUATIONS; WEAK SOLUTIONS; REGULARITY; DISSIPATION; EXISTENCE; CRITERION; FLOW;
D O I：
10.4171/AIHPC/129
中图分类号：
O29 [应用数学];
学科分类号：
070104 ;
摘要：
We address a degenerate elliptic variational problem arising in the application of the least action principle to averaged solutions of the inhomogeneous Euler equations in the Boussinesq approximation emanating from the horizontally flat Rayleigh-Taylor configuration. We give a detailed derivation of the functional starting from the differential inclusion associated with the Euler equations, i.e. the notion of an averaged solution is that of a subsolution in the context of convex integration, and illustrate how it is linked to the generalized least action principle introduced by Brenier [J. Amer. Math. Soc. 2 (1989), 225-255; in: New trends and results in mathematical description of fluid flows (2018), 53-75]. Concerning the investigation of the functional itself, we use a regular approximation in order to show the existence of a minimizer enjoying partial regularity, as well as other properties important for the construction of actual Euler solutions induced by the minimizer. Furthermore, we discuss to what extent such an application of the least action principle to subsolutions can serve as a selection criterion.
引用
收藏
页码：1527 / 1594
页数：68
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